# Jost Solution and the Spectrum of the Discrete Dirac Systems

- Elgiz Bairamov
^{1}Email author, - Yelda Aygar
^{1}and - Murat Olgun
^{1}

**2010**:306571

**DOI: **10.1155/2010/306571

© Elgiz Bairamov et al. 2010

**Received: **14 September 2010

**Accepted: **10 November 2010

**Published: **29 November 2010

## Abstract

We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.

## 1. Introduction

holds [1, chapter 3].

hold. The collection of quantities that specify to as the behaviour of the radial wave functions and at infinity is called the scattering of the BVP (1.1) and (1.2).

where and are real-valued continuous functions. In the case , , where is a potential function and the mass of a particle, (1.11) is called stationary Dirac system in relativistic quantum theory [5, chapter 7]. Jost solution and the scattering theory of (1.11) have been investigated in [6].

Jost solutions of quadratic pencil of Schrödinger, Klein-Gordon, and -Sturm-Liouville equations have been obtained in [7–9]. In [10–17], using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated.

Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problem (see the monographs of Agarwal [18], Agarwal and Wong [19], Kelley and Peterson [20], and the references therein). The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices [21].

In this paper, we find Jost solution of (1.12) and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that, , where denotes the continuous spectrum of , generated in by (1.12) and (1.13).

We also prove that under the condition (1.14) the operator has a finite number of simple real eigenvalues.

## 2. Jost Solution of (1.12)

where .

Theorem 2.1.

Proof.

By the condition (1.14), the series in the definition of ( ) are absolutely convergent. Therefore, ( ) can, by uniquely be defined by and , that is, the system (1.12) for and , has the solution given by (2.4) and (2.5).

where is the integer part of and is a constant. It follows from (2.4) and (2.11) that (2.3) holds.

Theorem 2.2.

The solution has an analytic continuation from to .

Proof.

From (1.14) and (2.11), we obtain that the series and are uniformly convergent in . This shows that the solution has an analytic continuation from to .

The functions and are called Jost solution and Jost function of the BVP (1.12) and (1.13), respectively. It follows from Theorem 2.2 that Jost solution and Jost function are analytic in and continuous on .

Theorem 2.3.

Proof.

From (2.15) and (2.16), we obtain (2.12).

## 3. Continuous and Discrete Spectrum of the BVP (1.12) and (1.13)

Theorem 3.1.

.

Proof.

where denotes the set of all eigenvalues of .

Definition 3.2.

The multiplicity of a zero of the function is called the multiplicity of the corresponding eigenvalue of .

Theorem 3.3.

Under the condition (1.14), the operator has a finite number of simple real eigenvalues.

Proof.

To prove the theorem, we have to show that the function has a finite number of simple zeros.

that is, all zeros of are simple.

holds, where .

There is a contradiction comparing (3.16) and (3.18). So and function has only a finite number of zeros.

## Authors’ Affiliations

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